homotopy T-algebra


Higher algebra

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




A homotopy TT-algebra over a Lawvere theory TT is a model for an \infty-algebra over TT, when the latter is regarded as an (∞,1)-algebraic theory.

As a model, homotopy TT-algebras are equivalent to strict simplicial algebras.


For TT (the syntactic category of) a Lawvere theory with generating object xx an ordinary algebra over a Lawvere theory functor TSetT \to Set that preserves products, in that for all nn \in \mathbb{N} the canonical morphism

i=1 nA(p i):A(x n)(A(x)) n \prod_{i = 1}^n A(p_i) : A(x^n) \to (A(x))^n

is an isomorphism.


A homotopy TT-algebra is a functor A:TA : T \to sSet with values in Kan complexes such that for all nn \in \mathbb{N} this canonical morphism is a weak homotopy equivalence.

For nn \in \mathbb{N} write F T(n)F_T(n) for the free simplicial TT-algebra on nn-generators, which is the image of x nx^n under the Yoneda embedding j:T op[T,sSet]j : T^{op} \to [T,sSet]. (See Lawvere theory for more on this.)


A homotopy TT-algebra is precisely


The fibrant objects in [T,sSet] proj[T,sSet]_{proj} are precisely the Kan complex-valued co-presheaves. Because F T(n)F_T(n) is representable, it is cofibrant in [T,sSet] proj[T,sSet]_{proj} (as one easily checks). Therefore the derived hom-spaces between F T()F_T(\cdots) and a degreewise Kan complex-valued AA may be computed simply as the sSet-hom-objects of the simplicial model category [T,sSet][T,sSet] and so the degreewise fibrant AA being a local object means that all morphisms of sSet-hom-objects

[T,sSet](F T(n),A)[T,sSet]( nF T(1),A). [T,sSet](F_T(n),A) \to [T,sSet](\coprod_n F_T(1), A) \,.

Due to the respect of the hom-functor for limits the expression on the right is

= n[T,sSet](F T(1),A). \cdots = \prod_n [T,sSet](F_T(1), A) \,.

Using the Yoneda lemma the morphism in question is indeed isomorphic to

A(x n)A(x) n. A(x^n) \to A(x)^n \,.

This observation motivated the following definition.


The model category structure for homotopy TT-algebras is the left Bousfield localization [T,sSet] proj,loc[T,sSet]_{proj,loc} of the projective model structure on simplicial presheaves [T,sSet] proj[T,sSet]_{proj} at the set of morphisms { nF T(1)F T(b)} n\{\coprod_n F_T(1) \to F_T(b)\}_{n \in \mathbb{N}}.



The model structure for homotopy TT-algebra [T,sSet] proj,loc[T,sSet]_{proj,loc} is a left proper simplicial model category.


Because the model structure on simplicial presheaves is and left Bousfield localization of model categories preserves these properties.


The inclusion

i:TAlg Δ op[T,sSet] i : T Alg^{\Delta^{op}} \hookrightarrow [T,sSet]

has a left adjoint

F:[T,sSet]TAlg Δ op F : [T,sSet] \to T Alg^{\Delta^{op}}

The limits in TAlgT Alg are easily seen to be limits in the underlying sets. Hence ii preserves all limits. The statement then follows by observing that the assumptions of the special adjoint functor theorem are met:

  • TAlgT Alg is complete;

  • it is a well powered category since [T,Set][T,Set] is and the subobject in TAlgT Alg are special subobjects in [T,Set][T,Set];

  • it has a small cogenerating set given by the representables.


An explicit description of FF is around HTT, lemma


Let TAlg proj Δ opT Alg^{\Delta^{op}}_{proj} be the category of simplcial T-algebras equipped with the standard model structure on simplicial algebras (with weak equivalences and fibrations the degreewise weak equivalences and fibrations in simplicial sets).

The adjunction from the previous lemma

TAlg Δ opF[T,sSet]=[T,Set] Δ op T Alg^{\Delta^{op}} \stackrel{\overset{F}{\leftarrow}}{\hookrightarrow} [T,sSet] = [T,Set]^{\Delta^{op}}

is a Quillen adjunction which is a Quillen equivalence

TAlg proj Δ op[T,sSet] proj,loc. T Alg^{\Delta^{op}}_{proj} \simeq [T,sSet]_{proj,loc} \,.

This is theorem 1.3 in (Badzioch)


The model structure on homotopy TT-algebras for T=T = CartSp the Lawvere theory of smooth algebras is considered in (Spivak) in the study of derived smooth manifold. (There is also a bit of disucssion of the relation to the model structure on simplicial algebras there.)



  • Bernard Badzioch, Algebraic theories in homotopy theory Annals of Mathematics, 155 (2002), 895-913 (JSTOR)

the model structure on homotopy TT-algebras is discussed and its Quillen equivalence to simplcial TT-algebras is proven.

A related discussion showing that simplicial TT algebras model all \infty-TT-algebras is in

  • Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.

The model structure on homotopy TT-algebras for T=T = CartSp the Lawvere theory of smooth algebras is considered in

Revised on November 25, 2010 00:26:13 by Urs Schreiber (